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Math Help - State the domain where the function is analytic

  1. #1
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    State the domain where the function is analytic

    State the domain where the function is analytic.
    f(z) = \frac{1}{1-z^4}

    From what I have studied, f(z) is analytic on a domain D, if f(z) is differentiable at all points in D.

    However, I don't know where I should begin. And I think that f(z) is not analytic at 1-z^4=0.

    Anyway, how to find the domain, how to show in details to solve this.

    Thank you.
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by noppawit View Post
    State the domain where the function is analytic.
    f(z) = \frac{1}{1-z^4}

    From what I have studied, f(z) is analytic on a domain D, if f(z) is differentiable at all points in D.

    However, I don't know where I should begin. And I think that f(z) is not analytic at 1-z^4=0.

    Anyway, how to find the domain, how to show in details to solve this.

    Thank you.
    As you have said, consider when 1-z^4=0.

    This would construct the 4th roots of unity z=i,\,z=-i,\,z=1,\,z=-1: f\!\left(z\right) is not differentiable at these values.

    Thus, our domain would be D=\mathbb{C}\backslash\{1,i,-1,-i\}.
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